We give polynomial-time constant-factor approximation algorithms
for the treewidth and branchwidth of any H-minor-free graph
for a given graph H with crossing number at most 1.
The approximation factors are 1.5 for treewidth and 2.25 for branchwidth.
In particular, our result directly applies to classes of nonplanar graphs
such as K5-minor-free graphs and
K3,3-minor-free graphs.
Along the way, we present a polynomial-time algorithm to decompose
H-minor-free graphs into planar graphs and graphs of treewidth at most
cH (a constant dependent on H)
using clique sums.
This result has several applications in designing fully polynomial-time
approximation schemes and fixed-parameter algorithms for many
NP-complete problems on these graphs.